MECHANICAL 1'H I I.i "Si > I'll Y. II VDROSTATICS. 



1.1 



slightly oscillate till the resistance of the fluid destroys 



iu motion, when it will firmly settle in a position of 



stable equilibrium with the diagonal A C vertical. (Fig. 



Fur it ia plain, that when this position is arrived 



Fit- 17*. 



A 



at, the centre of gravity G of the body, and the centre 

 of gravity g of the displaced fluid, are again in the same 

 vertical ; but the inertia of the rolling mass will cause 

 the axis C A to incline a little beyond the vertical posi- 

 tion ; the centre of gravity g of the displacement will 

 thus move a little towards the richt, as in the former 

 case ; and g being now nearer to G, the vertical form g 

 will meet the slightly inclined axis above G, so that the 

 new position of equilibrium into which the body finally 

 settles will be stable. 



In reference to the foregoing investigation, it may be 

 proper here to remark, that the student is not to infer 

 that the resulting determination of the metacentre is 

 merely a close approximation to its true position from 

 our having regarded as deprived of appreciable value. 

 The metacentre has been denned to be a point where cer- 

 tain two lines intersect : these two lines approach nearer 

 and nearer to coincidence as diminishes. Now if 0, 

 from any finite value, continually diminish, the point of 

 intersection referred to, will move along the axis, and 

 will cease to move and become stationary only when 

 becomes zero ; that is, when the two intersecting lines 

 actually coincide. It is this extreme limit of the inter- 

 sections that is, in strictness, the metacentre. It is 

 rigidly and accurately determined, from any general in- 

 vestigation in which is assumed to be a small value, 

 only in the extreme case of that hypothesis ; that is, 

 only when has diminished down to zero. It will be 

 seen, by re-examining the operations above, that the 

 final result strictly follows when becomes zero ; or 

 when is the extreme limit of the intersections of the 

 line of support with the axis H K. The metacentre, in 

 reference to a displacement in a given direction, is thus 

 a fixed point on the vertical, through the centre of 

 gravity ; and hag nothing to do with extent of dis- 

 turbance. When the body is at rest, the vertical 

 through the centre of gravity of the body, and that 

 through the centre of gravity of the fluid displaced, be- 

 come one and the same line : if the equilibrium be dis- 

 turbed, these verticals separate, and the point about which, 

 in this separation, they begin to turn, is the metacentre. 

 II. I ID- I-OMMI M. \TINi; Til IK ifii II I'.KNT 

 TUBES. In speaking of the Levelling Instrument at 

 page 761, we have regarded it as a vessel of fluid ; 

 T\t. l. and from the propo- 



sition which sug- 

 gested a reference to 

 it, have inferred that 

 the two surfaces of 

 the bent tube were 

 horizontal or level ; 

 but the following 

 direct proof that the 

 surfaces of the fluid 



iu any bent receptacle are necessarily horizontal, is de- 

 serving of notice for iti nimplicity. V> :i<>m 

 the Cvun Elrmcntairc dc Alfcanique of Do l.;iui>:iy. 



Let A li (Fig. 179) be two points taken in 1 1 

 of the fluid, and on the horizontal line wholly in the 

 channel of communication A It, between the two asct-ixl 

 ing portions of the bent tube ; these two portions being 

 of any relative bulk whatever. 



In order that the two points A and B may be at rest, 

 the pressures at A and B must be equal, and must be 

 the same as the pressure upon every point of the line 

 A B. Now the pressure on A is equal to the weight of 

 the vertical column of particles C A : the pressure on li, 

 on account of the obliquity of the arm B M, is not so 

 easily found. It may be obtained as follows : The 

 pressure on B is equal to the pressure on D, augmented 

 by the weight of the column of particles DK The 

 pressure on D is the same as that on E ; but the pres- 

 sure on E is equal to the pressure ouF, augmented by 

 the weight of the column of particles F E : consequently 

 the pressure on B is equal to the pressure on F, aug- 

 mented by the weight of the two columns of particles 

 D B, F E. Proceeding in this way, and observing that 

 the pressure on M is nothing, we see that the pressure 

 on B is equal to the weight of the five columns of par- 

 ticles DB, FE, HG, K I, M L. And since the pres- 

 sures on A and B are equal, it follows that the five 

 tical lines D B, F E, H G, K I, M L, are together equal 

 to the vertical line C A. Consequently every point in 

 the surfaces at C and M is at the same vertical distance 

 from the horizontal plane passing through the two points 

 A, B : these surfaces are therefore horizontal. 



In order that the foregoing phenomenon may always be 

 exhibited, it is of course necessary that the fluid intro- 

 duced into the tube be of uniform density, or that equal 

 vertical columns of it press with equal weights. Whin 

 two distinct fluids, which do not intermix, balance in a 

 bent tube, or in any two vessels having a channel of frc>o 

 communication with each other, the proportion becomes 

 modified, as follows : 



If two fluids which do not intermix are in equilibrio 

 in a bent tube, the heights of their surfaces above the 

 horizontal plane, where the one fluid rests upon the 

 other, are inversely as their special gravities. 

 Fig. 180. 



I'--- 1 



Let A B C be the bent tube, with a fluid A H of spe- 

 cific gravity S resting upon another fluid H B C of specific 

 gravity S', the horizontal plane of their separation being 

 at H. 



The pressure of the fluid A H on the horizontal plane 

 H is H X A E X Sg ; the pressure of the fluid C J, up- 

 ward on the same plane, is H X C F x &'J ; '" ' 

 plane H sustains only the upward pressure conmmnk . 

 by CJ, since, HJ being horizontal, if CJ wore re- 

 moved, there would be no upward pressure on 11. 



Hence, as the fluids are at rest, the pressures on H 

 must be equal 



/. AE xS-CFxS', 



that is, the heights of the surfaces A, 0, above the hori- 

 zontal plane of separation H J, are inversely as the spe- 

 cific gravities of the fluids AH, C J. 



If S' - S, we ih.-n hav.. th<- case of a single uniform 

 fluid before considered, and the result gives A E C F ; 



